Appearance of special functions of mathematical physics was associated with
solution of particular ordinary differential equations, while the integral
special functions arrived much later in mathematical literature after
properties of these functions were investigated. Integral special functions
were introduced as new special functions which can be applied in many
circumstances, especially in operational calculus, where they are frequently
serving as direct and inverse integral transforms. The form of integrand is
identical for all integral functions, but limits of integration are
different in order to assure the convergence of defined integrals. There are
two groups of integral special functions, these which have in integrands
elementary functions, and those having special functions. To the first group
belong the exponential integral \(-\mathrm {Ei}\left( -x\right) \) , the sine
and cosine integrals, \(\mathrm {si}\left( x\right) \) , \(\mathrm {Si}\left(x\right) \) , \(\mathrm {ci}(x)\)  and \(\mathrm {Ci}\left( x\right) \) , and the
corresponding integrals of hyperbolic trigonometric functions, \(\mathrm {Shi}\left( x\right) \)  and \(\mathrm {Chi}\left( x\right) \) . These functions are
defined in the following way [1]}, [2]}, [3]}, [4]}, [5]}
\(\begin{array}{l}\displaystyle \mathrm {E}_{1}\left( x\right) =-\mathrm {Ei}\left( -x\right)=\int _{x}^{\infty }\frac{e^{-t}}{t}dt,\quad x>0, \\\displaystyle \mathrm {Si}\left( x\right) =\int _{0}^{x}\frac{\sin t}{t}dt, \\\displaystyle \mathrm {si}\left( x\right) =-\int _{x}^{\infty }\frac{\sin t}{t}dt=\mathrm {Si}\left( x\right) -\frac{\pi }{2}, \\\displaystyle \mathrm {Ci}\left( x\right) =-\int _{x}^{\infty }\frac{\cos t}{t}dt=\gamma +\mathrm {\ln }x-\int _{0}^{x}\frac{1-\cos t}{t}dt=-\mathrm {ci}(x), \\\displaystyle \mathrm {Shi}\left( x\right) =\int _{0}^{x}\frac{\sinh t}{t}dt, \\\displaystyle \mathrm {Chi}\left( x\right) =\gamma +\mathrm {\ln }x-\int _{0}^{x}\frac{1-\cosh t}{t}dt,\end{array}\) 
